Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-3x+9y &= -3 \\ -2x-6y &= 8\end{align*}$
Explanation: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $2$ and the bottom equation by $3$ $\begin{align*}-6x+18y &= -6\\ -6x-18y &= 24\end{align*}$ Add the top and bottom equations. $-12x = 18$ Divide both sides by $-12$ and reduce as necessary. $x = -\dfrac{3}{2}$ Substitute $-\dfrac{3}{2}$ for $x$ in the top equation. $-3( -\dfrac{3}{2})+9y = -3$ $\dfrac{9}{2}+9y = -3$ $9y = -\dfrac{15}{2}$ $y = -\dfrac{5}{6}$ The solution is $\enspace x = -\dfrac{3}{2}, \enspace y = -\dfrac{5}{6}$.